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Reversed Field Pinch Approach to Magnetic Fusion S Reversed Field Pinch Approach to Magnetic Fusion S. Ortolani, Padova (Istituto Gas Ionizzati del C.N.R. Associazione Euratom-CNR) The process whereby a current chan­ Table 1 — RFP devices in operation, under construction and contemplated nel contracts under the influence of the magnetic field generated by the current ETA BETA II TPE-IR (M) ZT-40 (M) HBTX-IA OHTE RFX REACTOR (Padova, Italy) (Sakura-Mura, (Los Alamos, (Culham, UK) (General Atomic, (Padova, Italy) itself, is known as the "pinch effect" and Japan) USA) USA) it was first studied by Bennett in 1934. It Torus Major Radius, m 0.65 0.5 1.14 0.8 1.24 2 4.0 13 was also the first to be used experimen­ Torus Minor Radius, m 0.125 0.09 0.2 0.26 0.19 0.5 0.75 1.5 tally (by Cousins and Ware in 1951) to Plasma Current, MA 0.25 0.15 0.6 0.5 0.5 2 18.5 20 confine plasma in a magnetic field in First Operation 1979 1980 1981 1981 1981 1988 toroidal geometry. Indeed, during the 1950s pinches, both linear and toroidal, The first large toroidal device in which required for thermonuclear fusion (T were the most widely studied systems RFP configurations were studied was 10 keV, E 1 s) and to those achieved in fusion research. ZETA (UK), which operated during the in present day tokamaks (T 5 keV, E The main observation from these 1960s, where improved stability and = 0.1 s), but there is now a firmer scien­ early experiments was that the plasma confinement were observed when the tific basis for extending RFP research to was macroscopically unstable because plasma spontaneously generated a re­ bigger systems and higher currents. of the fast growing magnetohydrodyna- versed toroidal field in the outer region. While the scientific feasibility of con­ mic (MHD) instabilities which destroyed The importance of the reversed field trolled thermonuclear fusion will proba­ the symmetry of the plasma column and configuration both experimentally and bly be demonstrated in the tokamak led to wall contact. theoretically was better understood dur­ configurations 1), it is now recognized The addition of a longitudinal magne­ ing the 1970s. A number of small toroi­ that alternative schemes such as Mirrors tic field (Bz in cylindrical geometry, Fig. dal devices were built and operated in 2), Stellarators 3) and the RFP may offer 1; Bf in toroidal geometry, Fig. 2) was that period (UK, USA, Italy, Japan) and more advantageous solutions for nu­ suggested by theoretical studies and led many of the theoretical predictions of clear fusion reactors. to experiments on the so called "stabiliz­ the MHD theory were experimentally The RFP like the tokamak is a toroidal ed pinch" configuration (with Bf B ). confirmed. axisymmetric configuration but, where­ Although the macroscopic stability of During the same period, a major ad­ as in the tokamak the plasma current is the plasma improved, residual instabi­ vance in our theoretical understanding limited by stability requirements: lities and fluctuations caused poor of toroidal pinch discharges was made. / < 2πa2BfI o plasma confinement. It was realized that the most important the RFP can use high currents and One possible way of overcoming the property of the RFP is that the magnetic powerful ohmic heating to produce rela­ pinch instabilities, proposed during the configuration is a minimum energy state tively high-β ( 10%) plasmas without 1960s was the tokamak configuration in to which, under certain conditions, the the need for either very high toroidal which a large toroidal field is applied so plasma can spontaneously relax. This fields or powerful auxiliary heating that Bf/B is larger than the ratio of basic property was first discussed in methods. Furthermore it is now clear major to minor torus radius, R/a, and the 1974 by J.B. Taylor who revived Wolt- that although the RFP originally was most dangerous helical deformations of jer's astrophysical theorem on force-free thought to be intrinsically a pulsed the plasma cannot form because of the magnetic fields and extended it to labo­ device, the same processes which can periodicity conditions. At about the ratory plasma discharges. It is in practice lead to the formation of the configura­ same time a second approach appeared : this property of minimum energy state tion can also maintain it and give quasi the Reversed Field Pinch (RFP) in which which gives to the RFP its favourable steady state operation. Bf B but the toroidal field compo­ stability characteristics and makes it of nent in the outer region has the opposite particular interest among toroidal sys­ Equilibrium sign to that on the magnetic axis, thus tems for magnetic confinement. In the RFP the distributions of both the creating a high shear of the field lines A new generation of experiments has poloidal and toroidal field components and forming a stable radial distribution become operational during the last seve­ of the magnetic field. ral years (see Table 1) and it is now gene­ Fig. 1 —Cylindrical geometry. rally observed that the RFP configura­ Fig. 2 — Toroidal geometry. tion can confine relatively high β plas­ mas, heated by ohmic heating only, with moderate magnetic fields, Bf B < 1T. (P is the ratio between the plasma kinetic pressure and the magnetic field pressure. For a fusion reactor to be eco­ nomic it should be greater than about 10%.) The temperatures and confinement times achieved (T < 0.6 keV, E < 0.3 ms) are still modest compared to those 5 are mostly determined by currents flow­ sion and the hoop force of the toroidal ing in the plasma, the total current densi­ current. Equilibrium can be established ty being almost parallel to the magnetic through the influence of a conducting field. For an ideal equilibrium in toroidal wall or a vertical field. If the plasma is geometry (Fig. 2), the magnetic sur­ surrounded by a perfectly conducting faces are nested toroidal surfaces (Fig. shell, the equilibrium is achieved when 3) on which the magnetic field lines are the plasma shifts far enough that the in­ helically wound with a pitch length (for a creased magnetic pressure, due to the Fig. 3 — Toroidal magnetic Flux surFaces. circular cross section of the magnetic compression of poloidal flux, balances surfaces) : the outward force. In the RFP, because dients substantially smaller than the cri­ P = rBf/B = qR = f RI2π β is small and a/R can be small, only a tical value set by the Suydam's criterion. where f is the change in toroidal angle small outward shift (typically a few per­ of a magnetic line after traversing one cent of the plasma minor radius) is ne­ Minimum Energy State poloidal circuit and q represents the cessary to compress sufficient flux for Although stable high β equilibrium number of times the field line encircles toroidal equilibrium. profiles exist theoretically, they are dif­ the major axis in making one turn of the ficult to achieve experimentally by direct minor axis. It is worth noting that in a programming of the toroidal and poloidal tokamak, q > 1 (typically R/a) and in a Stability electrical circuits. The main reasons are RFP, q < 1 (typically < a/R). Moreover in Obviously a plasma confined by a technological : for a full control of the the RFP, the helical field lines have dif­ magnetic field away from the cold first profiles, programming on a short (MHD) ferent pitches on the various magnetic wall cannot be in the lowest energy state timescale implying high voltages, would surfaces (i.e. q is a function of minor as that would be characterized by ther­ be required. However these RFP confi­ radius) and the direction in the outer mal equilibrium. Confinement studies gurations have been demonstrated by region is opposite to that in the central aim then to find dynamical equilibria J.B. Taylor to be minimum energy states part of the toroidal plasma; this is illus­ with inhomogeneous plasma, and ma­ to which, under certain conditions, the trated in Fig. 4. gnetic field spatial distributions which plasma can spontaneously relax. can be stable for times longer than the RFPs can have a large aspect ratio R/a For a plasma surrounded by a perfect­ and therefore the radial pressure balance energy and particle confinement time. ly conducting wall, the minimum energy The spatial gradients of the plasma in the poloidal plane can be described in magnetic configuration is described by the simpler geometry of a straight kinetic pressure and the current density the solutions of the equation: cylinder (Fig. 1). In this case the equili­ distribution are the two main sources of x B = µB brium equation for the poloidal beta macroscopic instabilities. with p constant throughout the plasma. which is defined as: The stability for localised pressure This configuration corresponds however driven modes can be discussed on the to a large parallel current density flowing basis of Suydam's criterion. This states out to the wall. Also the z-component of that in cylindrical geometry, stability is the current density should substantially where p is the plasma pressure is : obtained when reverse in the outer region. Experimentally, the magnetic field dis­ tribution in the central region of the The two parameters F and which plasma is of the form given by Taylor's characterize the pinch are defined as : and shows that the destabilizing effect theory, but not in the outer region where F = Bz(a) <Bz> and = B (a)/<Bz>. of the negative pressure gradient can be there is little or no current because the These equations reduce with βf = 1 to compensated by a sufficiently large plasma temperature near the wall is the Bennett equilibrium equation: shear (dq/dr) in the magnetic field.
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